Practical Ship Hydrodynamics

Introduction 23
experts, while boundary element methods are more widely used. For further
studies, the book by Ferziger and Peric is recommended.
1.5.1 Turbulence models
The RANSE equations require external turbulence models to couple the
Reynolds stresses (terms from the turbulent fluctuations) to the time-averaged
velocities. Turbulence is in general not fully understood. All turbulence
models used for ship flows are semi-empirical. They use some theories about
the physics of turbulence and supply the missing information by empirical
constants. None of the turbulence models used so far for ship flows has been
investigated for its suitability at the free surface. On the other hand, it is
not clear whether an exact turbulence modelling in the whole fluid domain
is necessary for engineering purposes. There are whole books on turbulence
models and we will discuss here only the most primitive turbulence models
which were most popular in the 1990s, especially as they were standard
options in commercial RANSE solvers. ITTC (1990) gives a literature review
of turbulence models as applied to ship flows.
Turbulence models may be either algebraic (0-equation models) or based
on one or more differential equations (1-equation models, 2-equation models
etc.). Algebraic models compute the Reynolds stresses directly by an algebraic
expression. The other models require the parallel solution of additional
differential equations which is more time consuming, but (hopefully) also
more accurate.
The six Reynolds stresses (or more precisely their derivatives) introduce six
further unknowns. Traditionally, the Boussinesq approach has been used in
practice which assumes isotropic turbulence, i.e. the turbulence properties are
independent of the spatial direction. (Detailed measurements of ship models
have shown that this is not true in some critical areas in the aftbody of full
ships. It is unclear how the assumption of isotropic turbulence affects global
properties like the wake in the propeller plane.) The Boussinesq approach then
couples the Reynolds stresses to the gradient of the average velocities by an
eddy viscosity t:
�
u0u0 v0u0 w0u0 � � �
2ux uy C vx uz C wx
u 0 v 0 v 0 v 0 w 0 v 0
u 0 w 0 v 0 w 0 w 0 w 0
D t
⎡
⎣
uy C vx 2vy wy C vz
uz C wx wy C vz 2wz
2
⎤
3 k 0 0
2
0 3 k 0 ⎦
0 0 2
3 k
k is the (average) kinetic energy of the turbulence:
k D 1
2 ⊲u2 C v 2 C w 2 ⊳
The eddy viscosity t has the same dimension as the real viscosity , but
unlike it is not a constant, but a scalar depending on the velocity field. The
eddy viscosity approach transforms the RANSE to:
⊲ut C uux C vuy C wuz⊳ D f1 px
2
3 kx C ⊲ C t⊳⊲uxx C uyy C uzz⊳
C tx2ux C ty⊲uy C vx⊳ C tz⊲uz C wx⊳